# Eisenstein prime

Given the complex cubic root of unity^{} $\omega ={e}^{\frac{2i\pi}{3}}$, an Eisenstein integer^{} $a\omega +b$ (where $a$ and $b$ are natural integers) is said to be an *Eisenstein prime ^{}* if its only divisors

^{}are 1, $\omega $, $1+\omega $ and itself.

Eisenstein primes of the form $0\omega +b$ are ordinary natural primes $p\equiv 2mod3$. Therefore no Mersenne prime^{} is also an Eisenstein prime.

Title | Eisenstein prime |
---|---|

Canonical name | EisensteinPrime |

Date of creation | 2013-03-22 16:10:10 |

Last modified on | 2013-03-22 16:10:10 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11R04 |

Related topic | EisensteinIntegers |